9 FURTHER APPLICATIONS OF INTEGRATION

9.5 PROBABILITY
TRANSPARENCY AVAILABLE
#18 (Figure 5)
SUGGESTED TIME AND EMPHASIS
1 class Optional material
POINTS TO STRESS
1. The definition of a probability density function, and how it can be used to determine P (a ≤ x ≤ b).
2. The definition of a mean, algebraically and geometrically.
3. The uses of the normal distribution.
QUIZ QUESTIONS
• Text Question: Estimate the probability that a North American male is precisely six feet tall.
Answer: 0
• Drill Question: Assume that the probability density function of a continuous random variable X is given
1
by f (X) =
π ( 1 + X 2). What is the probability that 0 < X < √ 3?
Answer: 1 π
∫ √ 3
0
dx
1 + x 2 = 1 3
MATERIALS FOR LECTURE
• Describe Buffon’s needle problem as follows (see Problem 11 in Problems Plus after this chapter):
In a famous 18th-century problem, known as Buffon’s needle problem, a needle of length h is dropped
onto a flat surface (for example, a table) on which parallel lines L units apart, L ≥ h, have been drawn.
The problem is to determine the probability that the needle will come to rest intersecting one of the lines.
Assume that the lines run east-west, parallel to the x-axis in a rectangular coordinate system (as in the
figure below).
L
y
h
¬
h sin ¬
y
L
h
y=h sin ¬
¹ _2 ¹
¬
Let y be the distance from the “southern” end of the needle to the nearest line to the north. (If the needle’s
southern end lies on a line, let y = 0. If the needle happens to lie precisely east-west, let the “western”
end be the “southern” end.) Then 0 ≤ y ≤ L and 0 ≤ θ ≤ π. Note that the needle intersects one of the
lines only when y < h sin θ. Now, the total set of possibilities for the needle can be identified with the
rectangular region 0 ≤ y ≤ L, 0≤ θ ≤ π, and the proportion of times that the needle intersects a line is
511